# Predicate logic

**Predicate logic** is formal logic where the alphabet includes quantifiers (, …), variables (*x*, *y*, *z*…), and **predicate symbols** (*P*, *Q*…) that can relate variables or constant terms.

For example, is a sentence in predicate logic, meaning: “for all *x*, there exists some *y*, such that either P(*x*, *y*) holds, or Q(*y*, *x*) does not hold.”

In this case, P and Q are “non-logical symbols” standing for some relations, and x and y are variables ranging over some implicit domain. The precise relations and domain are opaque to the *rules* of predicate logic — the point of formal logic is arguably to state inferences like without worrying about what *x* and *P* stand for — but can be assigned meaning when combined with some **interpretation** or **model** for the logic.

Even in its simplest form, predicate logic can be used to model many statements in natural language, where the predicate symbols and constants/terms are taken to model verbs relating nouns:

Every person who lives in Perth lives in Australia.

Toaq is based on this formalism. Its content words are explicitly *n*-ary predicates:

poq: ___ is a person.

bua: ___ lives in ___.

It allows for slots that themselves accept other predicates/relations, meaning Toaq is really based on **higher-order logic**:

cheo: ___ all satisfy binary relation ___ among each other.

This connection between predicate logic and natural language has been explored since the 1970s by Montague (*Montague grammar* on Wikipedia).

## See also

- Predicate logic on Wikipedia.
- Plural logic, another flavor of the logic Toaq is based on.